7. Systems of Equations & Matrices

Determinants and Cramer's Rule

7. Systems of Equations & Matrices

# Determinants and Cramer's Rule - Video Tutorials & Practice Problems

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concept

## Determinants of 2×2 Matrices

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4mPlay a video:

2

Problem

ProblemEvaluate the determinant of the matrix**.**

A

$\frac{107}{3}$

B

$\frac{227}{6}$

C

33

D

12

3

concept

## Cramer's Rule - 2 Equations with 2 Unknowns

Video duration:

6mPlay a video:

4

Problem

ProblemWrite each equation in standard form and use Cramer's Rule to solve the system**.**

$y=-3x+4$

$-2x=7y-9$

A

$x=1,y=1$

B

$x=-1,y=1$

C

$x=1,y=-1$

D

$x=-1,y=-1$

5

Problem

ProblemWrite each equation in standard form and use Cramer's Rule to solve the system**.**

$y-9x=-3$

$-3x=4y-1$

A

$y=3,x=0$

B

$x=0,y=3$

C

$x=-\frac13,y=1$

D

$x=\frac13,y=0$

6

concept

## Determinants of 3×3 Matrices

Video duration:

7mPlay a video:

7

Problem

ProblemEvaluate the determinant of the matrix**.**

****

A

165

B

9

C

63

D

25

8

concept

## Cramer's Rule - 3 Equations w/ 3 Unknowns

Video duration:

13mPlay a video:

9

Problem

ProblemSolve the system of equations using Cramer's Rule**.**

$4x+2y+3z=6$

$x+y+z=3$

$5x+y+2z=5$

A

$x=-2,y=-8,z=4$

B

$x=1,y=4,z=-2$

C

$x=2,y=8,z=-4$

D

$x=-1,y=-4,z=2$

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PRACTICE PROBLEMS AND ACTIVITIES (81)

- What is the value of ?
- Evaluate each determinant in Exercises 1–10. 5 7 2 3
- What expression in x represents ?
- Evaluate each determinant in Exercises 1–10. - 4 1 5 6
- Evaluate each determinant in Exercises 1–10. - 7 14 2 - 4
- What is the value of x if = 9?
- Evaluate each determinant. See Example 1.
- Evaluate each determinant in Exercises 1–10. - 5 - 1 - 2 - 7
- Evaluate each determinant in Exercises 1–10. - 5 - 1 - 2 - 7
- Evaluate each determinant. See Example 1.
- Evaluate each determinant in Exercises 1–10. 1/2 1/2 1/8 - 3/4
- For Exercises 11–22, use Cramer's Rule to solve each system. x + y = 7 x - y = 3
- Evaluate each determinant. See Example 1.
- Evaluate each determinant. See Example 1.
- For Exercises 11–22, use Cramer's Rule to solve each system. 12x + 3y = 15 2x - 3y = 13
- For Exercises 11–22, use Cramer's Rule to solve each system. 12x + 3y = 15 2x - 3y = 13
- Evaluate each determinant. See Example 1.
- For Exercises 11–22, use Cramer's Rule to solve each system. 4x - 5y = 17 2x + 3y = 3
- For Exercises 11–22, use Cramer's Rule to solve each system. x + 2y = 3 3x - 4y = 4
- Find the cofactor of each element in the second row of each matrix. See Example 2.
- Find the cofactor of each element in the second row of each matrix. See Example 2.
- For Exercises 11–22, use Cramer's Rule to solve each system. 3x - 4y = 4 2x + 2y = 12
- For Exercises 11–22, use Cramer's Rule to solve each system. 3x - 4y = 4 2x + 2y = 12
- Evaluate each determinant. See Example 3.
- For Exercises 11–22, use Cramer's Rule to solve each system. 2x = 3y + 2 5x = 51 - 4y
- In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 0 0 2 1 - 5 2 5 - 1
- Evaluate each determinant. See Example 3.
- Evaluate each determinant. See Example 3.
- In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 1 0 - 3 4 0 - 1 3 - 5
- In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 1 0 - 3 4 0 - 1 3 - 5
- Evaluate each determinant. See Example 3.
- In Exercises 23–30, use expansion by minors to evaluate each determinant. 1 1 1 2 2 2 - 3 4 - 5
- In Exercises 23–30, use expansion by minors to evaluate each determinant. 0.5 7 5 0.5 3 9 0.5 1 3
- In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate eac...
- In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate eac...
- In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate eac...
- Evaluate each determinant.
- In Exercises 37–44, use Cramer's Rule to solve each system. x + y + z = 0 2x - y + z = - 1 - x + 3y - z = - ...
- In Exercises 37–44, use Cramer's Rule to solve each system. x + y + z = 0 2x - y + z = - 1 - x + 3y - z = - ...
- Evaluate each determinant.
- In Exercises 37–44, use Cramer's Rule to solve each system. 4x - 5y - 6z = - 1 x - 2y - 5z = - 12 2x - y = 7
- Evaluate each determinant.
- In Exercises 37–44, use Cramer's Rule to solve each system. x + y + z = 4 x - 2y + z = 7 x + 3y + 2z = 4
- In Exercises 37–44, use Cramer's Rule to solve each system. x + 2z = 10 2y - z = - 5 2x + 3y = 13
- In Exercises 37–44, use Cramer's Rule to solve each system. x + 2z = 10 2y - z = - 5 2x + 3y = 13
- In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussia...
- In Exercises 46–51, evaluate each determinant.
- In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussia...
- Use the determinant theorems to evaluate each determinant. See Example 4.
- Solve each equation. = 8
- Solve each equation. = 2x
- Evaluate each determinant in Exercises 49–52. 4 2 8 - 7 - 2 0 4 1 5 0 0 5 4 0 0 - 1
- In Exercises 46–51, evaluate each determinant.
- Evaluate each determinant in Exercises 49–52. 4 2 8 - 7 - 2 0 4 1 5 0 0 5 4 0 0 - 1
- Use the determinant theorems to evaluate each determinant. See Example 4.
- Evaluate each determinant in Exercises 49–52. - 2 - 3 3 5 1 - 4 0 0 1 2 2 - 3 2 0 1 1
- In Exercises 46–51, evaluate each determinant.
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 46–51, evaluate each determinant.
- In Exercises 53–54, evaluate each determinant. | | 3 1| |7 0| | | |- 2 3| |1 5| | | | | | 3 0| |9 - 6| | | |...
- In Exercises 52–55, use Cramer's Rule to solve each system.
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 52–55, use Cramer's Rule to solve each system.
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants...
- In Exercises 57–60, solve each equation for x. |- 2 x| | | = 32 | 4 6|
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 57–60, solve each equation for x. |1 x - 2| |3 1 1| = - 8 |0 - 2 2|
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 57–60, solve each equation for x. |1 x - 2| |3 1 1| = - 8 |0 - 2 2|
- Use the determinant theorems to evaluate each determinant. See Example 4.
- Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the soluti...
- Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the soluti...
- Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the soluti...